Abstract: With the presence of frictions and constraints of ii=0, it is difficult to simulate a plastic forming process involving frictional contacts by using a conventional method such as finite element analysis. However, it is shown that simulations can be performed by employing the incomplete variational inequality principle. Variational inequalities are established by introducing the friction constraints into the energy functional, allowing changes in material volume and dealing with the volume strain by means of penalty factor. Example calculations indicate that, by using the incomplete variational inequality principle, not only the velocity field but also the deformation force can be predicted. The predicted deformation force is found to be closer to that from measurement than the solution produced from the sliding line field combined with the optimization method. The incomplete variational inequality principle can generate accurate results even in the case of finite deformation. Variational inequalities derived from the incomplete variational inequality principle are highly mathematically complicated, therefore, the particle swarm optimization (PSO) method is explored in order to solve these inequalities. The randomness of solutions from PSO is depressed by performing an average process. Not only the solution process becomes more stable, but also the results are improved. A new application of the PSO is thus demonstrated.

Key words: Finite deformation, Frictional constraint, Particle swatm optimization (PSO), Rigid plasticity, Variational inequalities principles, “S” Specimen, Error Tracing, Fuzzy Comprehensive Evaluation, Mapping Relationship, Key words: Five-axis CNC Machine Tool